The question arises (of interest from a theoretical point of view, though irrelevant for applications as in [1]} or [2]}) what precisely happens at the thresholds. The first part of the following is concerned with this question. It turns out that here a monotonicity property of the quotient \({\mathbb {E}}\,f_k(C_N)/\binom{N}{k}\)  as a function of \(N\)  is useful, and this will be proved in Theorem REF . This monotonicity may be of independent interest. Here it allows us to give an elementary proof that the limit in (REF ) is equal to 1 also for \(\delta =\frac{1}{2}\) . Curiously, none of the proofs given in [3]} for \(\delta <1/2\)  or \(\delta >1/2\)  extends to include the case \(\delta =1/2\) , so a different argument is required. A proof using deeper deviation results from probability theory was given by Godland, Kabluchko and Thäle [4]}.
